# Questions tagged [matching]

A matching (aka **Independent Edge Set**) in a simple graph is the set of pairwise non-adjacent edges i.e. no two edges have common vertex.

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### Comparing two poses of a point cloud

I have an unstructured point cloud for which I determine pose information described as a similarity transform (rotation matrix and translation vector). In my setting, several independent poses are ...
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1 vote
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### Max matching algorithm lemma approximation algorithm

We have this algorithm which is supposed to find max matchings. ...
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### Proving that the number of leaves in a tree >= number of unmatched vertices

Consider a rooted tree $T$. A matching in $T$ is said to be proper if for every unmatched vertex $v$ it holds that the parent of $v$ is matched to one of the siblings of $v$. It is known that every ...
• 379
1 vote
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### Is Set Cover problem with subsets of size ≤2 solvable in polynomial time?

I came across the below question where the polynomial time solution to the "Set Cover Problem" is discussed when the subsets are of size EXACTLY 2. Set cover problem with sets of size 2 The ...
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### Matching problem in bipartite network with more than one edge per vertex

I'm interested to know if there is an algorithm to find possible solutions for the matching problem, in a bipartite network where each vertex have maximum number of connections greater than one. For ...
• 123
1 vote
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### Find a perfect matching with weights as close as possible to each other

Given a set of jobs $J$ and a set of machines $M$, where the link between machine $i\in M$ and job $j\in J$ has a positive weight $w_{ij}$. The problem is to select a perfect matching between the jobs ...
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### Using algorithm for weighted graphs when the weights are vectors

Consider the following example problem. Given a graph with edge weights, find a matching that maximizes the number of matched vertices, and subject to this, maximizes the total weight. This problem ...
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### Proving existence of sinkless orientation on graph with minimum degree 2

I am given a graph of minimum degree at least 2 (not necessairly regular). I want to prove that there is a way to orient the edges of G such that each node of G has at least one out-going edge. As a ...
• 149
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### Prove there is a matching of size n/2 on a graph with 2n vertices each of degree n

Given underirected $n$-regular graph with $2n$ nodes, I am asked to show it has a matching of size $n/2$. My attempt: At each step I will also remove the edge from the graph that I am adding to the ...
• 149
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### Find a maximum matching in linear time - How to prove it?

Let B be an undirected tree with |V| nodes given as adjacency list. I want to develop a greedy algorithm using pseudo code to find a maximal matching in runtime O(|V|). I understand that the solution ...
1 vote
26 views

### Max-Min Weighted Matching

The maximum weighted matching problem (https://en.wikipedia.org/wiki/Maximum_weight_matching) finds a matching in a weighted graph that has maximum sum of weights. I was wondering if there are any ...
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1 vote
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### Graph in which greedy algorithm for maximum matching is a 2-approximation

Here is a greedy algorithm for maximum bipartite matching: Iteratively select an edge that is not incident to previously selected edges. This algorithm returns a 2-approximation, and runs in linear ...
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### Find optimal local alignment of two strings with restrictions

I have a homework question that I trying to solve for many hours without success, maybe someone can guide me to the right way of thinking about it. The problem: We want to find an optimal local ...
• 165
1 vote
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### Concept of M-augmenting path to find a larger matching than $M$

I'm reading section 16.1 of the book, Combinatorial optimization, Polyhedra and efficiency by Schrijver. Here, he starts with a matching $M$ and describes a path $P$ that is $M$-augmenting if: The ...
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1 vote
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### splitting of people betwen groups with group prioritiy list per person

Looking for an algorithm for splitting a list of people (S) into groups (G), |S| <= |G|, more than one person wants to be in a certain group. Every person has a monotone ranking from highest (most ...
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### How difficult is this matching-like problem?

Let $A$ and $B$ be two sets of integers with $|A|>|B|$. Given a map $f: A \rightarrow B$ and $i \in A, j \in B$, let us use the shorthand "$i$ is matched to $j$" if $f(i)=j$. I am ...
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### Game on the graph with matchings

The game on the graph $G$ is defined as follows. Initially, the chip is located at one of the vertices (let's call it the starting one). The players take turns, on each move it is necessary to move ...
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### Maximal vs maximum matchings

Let $M_1$ be an inclusion-maximal matching in $G$ (that is, there is no matching which strictly contains it), and $M_2$ a maximum-size matching in $G$. How to prove that $|M_2| \le 2|M_1|$?
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1 vote
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### Independent sets into which all the vertices of the graph can be split

How to prove that if $G$ is an acyclic transitive digraph, then the least independent sets into which all vertices of G can be divided is equal to the size of the longest paths to $G$?
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